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Sep 23, 2013
09/13

by
Yann Bugeaud

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We establish new combinatorial transcendence criteria for continued fraction expansions. Let $\alpha = [0; a_1, a_2,...]$ be an algebraic number of degree at least three. One of our criteria implies that the sequence of partial quotients $(a_{\ell})_{\ell \ge 1}$ of $\alpha$ cannot be generated by a finite automaton, and that the complexity function of $(a_{\ell})_{\ell \ge 1}$ cannot increase too slowly.

Source: http://arxiv.org/abs/1012.1709v2

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11

Sep 19, 2013
09/13

by
Yann Bugeaud; Nikolay Moshchevitin

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We prove that there exist arbitrarily small positive real numbers $\epsilon$ such that every integral power $(1 + \vepsilon)^n$ is at a distance greater than $2^{-17} \epsilon |\log \vepsilon|^{-1}$ to the set of rational integers. This is sharp up to the factor $2^{-17} |\log \epsilon|^{-1}$. We also establish that the set of real numbers $\alpha > 1$ such that the sequence of fractional parts $(\{\alpha^n\})_{n \ge 1}$ is not dense modulo 1 has full Hausdorff dimension.

Source: http://arxiv.org/abs/1009.4528v1

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6.0

Sep 18, 2013
09/13

by
Yann Bugeaud; Aleksandar Ivić

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Sums of the form $\sum_{n\le x}E^k(n) (k\in{\bf N}$ fixed) are investigated, where $$ E(T) = \int_0^T|\zeta(1/2+it)|^2 dt - T\Bigl(\log {T\over2\pi} + 2\gamma -1\Bigr)$$ is the error term in the mean square formula for $|\zeta(1/2+it)|$. The emphasis is on the case k=1, which is more difficult than the corresponding sum for the divisor problem. The analysis requires bounds for the irrationality measure of ${\rm e}^{2\pi m}$ and for the partial quotients in its continued fraction expansion.

Source: http://arxiv.org/abs/0707.4275v1

7
7.0

Sep 23, 2013
09/13

by
Boris Adamczewski; Yann Bugeaud

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We use the Schmidt Subspace Theorem to establish the transcendence of a class of quasi-periodic continued fractions. This improves earlier works of Maillet and of A. Baker. We also improve an old result of Davenport and Roth on the rate of increase of the denominators of the convergents to any real algebraic number.

Source: http://arxiv.org/abs/math/0511681v1

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Sep 23, 2013
09/13

by
Boris Adamczewski; Yann Bugeaud

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The continued fraction expansion of an irrational number $\alpha$ is eventually periodic if and only if $\alpha$ is a quadratic irrationality. However, very little is known regarding the size of the partial quotients of algebraic real numbers of degree at least three. Because of some numerical evidence and a belief that these numbers behave like most numbers in this respect, it is often conjectured that their partial quotients form an unbounded sequence. More modestly, we may expect that if the...

Source: http://arxiv.org/abs/math/0511677v1

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15

Sep 23, 2013
09/13

by
Boris Adamczewski; Yann Bugeaud

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For any given real number $\alpha$ with bounded partial quotients, we construct explicitly continuum many real numbers $\beta$ with bounded partial quotients for which the pair $(\alpha, \beta)$ satisfies a strong form of the Littlewood conjecture. Our proof is elementary and rests on the basic theory of continued fractions.

Source: http://arxiv.org/abs/math/0511678v1

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10.0

Sep 23, 2013
09/13

by
Boris Adamczewski; Yann Bugeaud

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Let $b \ge 2$ be an integer. We prove that the $b$-adic expansion of every irrational algebraic number cannot have low complexity. Furthermore, we establish that irrational morphic numbers are transcendental, for a wide class of morphisms. In particular, irrational automatic numbers are transcendental. Our main tool is a new, combinatorial transcendence criterion.

Source: http://arxiv.org/abs/math/0511674v1

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Jul 20, 2013
07/13

by
Yann Bugeaud; Andrej Dujella

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We establish new results on root separation of integer, irreducible polynomials of degree at least four. These improve earlier bounds of Bugeaud and Mignotte (for even degree) and of Beresnevich, Bernik, and Goetze (for odd degree).

Source: http://arxiv.org/abs/1007.3406v2

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Sep 21, 2013
09/13

by
Yann Bugeaud; Simon Kristensen

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We are studying the Diophantine exponent \mu_{n,l}$ defined for integers 1 \leq l < n and a vector \alpha \in \mathbb{R}^n by letting \mu_{n,l} = \sup{\mu \geq 0: 0 < ||x \cdot \alpha|| < H(x)^{-\mu} for infinitely many x \in C_{n,l} \cap \mathbb{Z}^n}, where \cdot is the scalar product and || . || denotes the distance to the nearest integer and C_{n,l} is the generalised cone consisting of all vectors with the height attained among the first l coordinates. We show that the...

Source: http://arxiv.org/abs/0709.0854v1

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Jul 20, 2013
07/13

by
Yann Bugeaud; Michel Laurent

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Let x be a real number and let n be a positive integer. We define four exponents of Diophantine approximation, which complement the exponents w_n(x) and w_n^*(x) defined by Mahler and Koksma. We calculate their six values when n=2 and x is a real number whose continued fraction expansion coincides with some Sturmian sequence of positive integers, up to the initial terms. In particular, we obtain the exact exponent of approximation to such a continued fraction x by quadratic surds.

Source: http://arxiv.org/abs/math/0406064v1

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Sep 23, 2013
09/13

by
Boris Adamczewski; Yann Bugeaud

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Let $\k$ be an arbitrary field. For any fixed badly approximable power series $\Theta$ in $\k((X^{-1}))$, we give an explicit construction of continuum many badly approximable power series $\Phi$ for which the pair $(\Theta, \Phi)$ satisfies the Littlewood conjecture. We further discuss the Littlewood conjecture for pairs of algebraic power series.

Source: http://arxiv.org/abs/math/0511680v1

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Sep 18, 2013
09/13

by
Yann Bugeaud; Michel Laurent

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The paper is mostly a survey on recent results in Diophantine approximation, with emphasis on properties of exponents measuring various notions of Diophantine

Source: http://arxiv.org/abs/math/0611354v1

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7.0

Sep 21, 2013
09/13

by
Yann Bugeaud; Nikolay Moshchevitin

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We establish that the set of pairs $(\alpha, \beta)$ of real numbers such that $$ \liminf_{q \to + \infty} q \cdot (\log q)^2 \cdot \Vert q \alpha \Vert \cdot \Vert q \beta \Vert > 0, $$ where $\Vert \cdot \Vert$ denotes the distance to the nearest integer, has full Hausdorff dimension in $\R^2$. Our proof rests on a method introduced by Peres and Schlag, that we further apply to various Littlewood-type problems

Source: http://arxiv.org/abs/0905.0830v1

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Sep 18, 2013
09/13

by
Yann Bugeaud; Florian Luca

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In this paper, we prove that the period of the continued fraction expansion of ${\sqrt {2^{n}+1}}$ tends to infinity when $n$ tends to infinity through odd positive integers.

Source: http://arxiv.org/abs/math/0401341v1

8
8.0

Jul 20, 2013
07/13

by
Yann Bugeaud; Michel Laurent

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In Diophantine approximation, inhomogeneous problems are linked with homogeneous ones by means of the so-called Transference Theorems. We revisit this classical topic by introducing new exponents of Diophantine approximation. We prove that the exponent of approximation to a generic point in R^n by a system of n linear forms is equal to the inverse of the uniform homogeneous exponent associated to the system of dual forms.

Source: http://arxiv.org/abs/math/0406065v1

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Sep 23, 2013
09/13

by
Yann Bugeaud; Bernard De Mathan

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Let $d$ be a positive integer. Let $p$ be a prime number. Let $\alpha$ be a real algebraic number of degree $d+1$. We establish that there exist a positive constant $c$ and infinitely many algebraic numbers $\xi$ of degree $d$ such that $|\alpha - \xi| \cdot \min\{|\Norm(\xi)|_p,1\} < c H(\xi)^{-d-1} (\log 3 H(\xi))^{-1/d}$. Here, $H(\xi)$ and $\Norm(\xi)$ denote the na{\"\i}ve height of $\xi$ and its norm, respectively. This extends an earlier result of de Mathan and Teuli\'e that...

Source: http://arxiv.org/abs/0903.2741v1

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15

Sep 21, 2013
09/13

by
Yann Bugeaud; Jan-Hendrik Evertse

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we derive new, improved lower bounds for the block complexity of an irrational algebraic number and for the number of digit changes in the b-ary expansion of an irrational algebraic number. To this end, we apply a quantitative version of the Subspace Theorem due to Evertse and Schlickewei (2002).

Source: http://arxiv.org/abs/0709.1560v1

8
8.0

Sep 21, 2013
09/13

by
Yann Bugeaud; Jan-Hendrik Evertse

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We investigate how well complex algebraic numbers can be approximated by algebraic numbers of degree at most n. We also investigate how well complex algebraic numbers can be approximated by algebraic integers of degree at most n+1. It follows from our investigations that for every positive integer n there are complex algebraic numbers of degree larger than n that are better approximable by algebraic numbers of degree at most n than almost all complex numbers. As it turns out, these numbers are...

Source: http://arxiv.org/abs/0709.1556v1

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8.0

Sep 22, 2013
09/13

by
Yann Bugeaud; Maurice Mignotte; Samir Siksek

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This is the first in a series of papers whereby we combine the classical approach to exponential Diophantine equations (linear forms in logarithms, Thue equations, etc.) with a modular approach based on some of the ideas of the proof of Fermat's Last Theorem. In this paper we give new improved bounds for linear forms in three logarithms. We also apply a combination of classical techniques with the modular approach to show that the only perfect powers in the Fibonacci sequence are 0, 1, 8, 144...

Source: http://arxiv.org/abs/math/0403046v1

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6.0

Sep 21, 2013
09/13

by
Yann Bugeaud; Dalia Krieger; Jeffrey Shallit

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Let $\mb w$ be a morphic word over a finite alphabet $\Sigma$, and let $\Delta$ be a nonempty subset of $\Sigma$. We study the behavior of maximal blocks consisting only of letters from $\Delta$ in $\mb w$, and prove the following: let $(i_k,j_k)$ denote the starting and ending positions, respectively, of the $k$'th maximal $\Delta$-block in $\mb w$. Then $\limsup_{k\to\infty} (j_k/i_k)$ is algebraic if $\mb w$ is morphic, and rational if $\mb w$ is automatic. As a result, we show that the same...

Source: http://arxiv.org/abs/0808.2544v2

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9.0

Sep 20, 2013
09/13

by
Yann Bugeaud; Maurice Mignotte; Samir Siksek

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We solve completely the Lebesgue-Nagell equation x^2+D=y^n, in integers x, y, n>2, for D in the range 1 =< D =< 100.

Source: http://arxiv.org/abs/math/0405220v1

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7.0

Sep 19, 2013
09/13

by
Yann Bugeaud; Alan Haynes; Sanju Velani

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The main goal of this note is to develop a metrical theory of Diophantine approximation within the framework of the de Mathan-Teulie Conjecture, also known as the `Mixed Littlewood Conjecture'. Let p be a prime. A consequence of our main result is that, for almost every real number \alpha, \liminf_{n\rar\infty}n(\log n)^2|n|_p\|n\alpha\|=0.

Source: http://arxiv.org/abs/0909.3923v2

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175

Jul 20, 2013
07/13

by
Yann Bugeaud; Pascal Hubert; Thomas A. Schmidt

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We give the first transcendence results for the Rosen continued fractions. Introduced over half a century ago, these fractions expand real numbers in terms of certain algebraic numbers.

Source: http://arxiv.org/abs/1007.2050v1

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Sep 19, 2013
09/13

by
Ryan Broderick; Yann Bugeaud; Lior Fishman; Dmitry Kleinbock; Barak Weiss

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Given b > 1 and y \in \mathbb{R}/\mathbb{Z}$, we consider the set of $x\in \mathbb{R}$ such that $y$ is not a limit point of the sequence $\{b^n x \bmod 1: n\in\N\}$. Such sets are known to have full Hausdorff dimension, and in many cases have been shown to have a stronger property of being winning in the sense of Schmidt. In this paper, by utilizing Schmidt games, we prove that these sets and their bi-Lipschitz images must intersect with `sufficiently regular' fractals $K\subset...

Source: http://arxiv.org/abs/0909.4251v3

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Sep 23, 2013
09/13

by
Boris Adamczewski; Yann Bugeaud; Les J. L. Davison

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It is widely believed that the continued fraction expansion of every irrational algebraic number $\alpha$ either is eventually periodic (and we know that this is the case if and only if $\alpha$ is a quadratic irrational), or it contains arbitrarily large partial quotients. Apparently, this question was first considered by Khintchine. A preliminary step towards its resolution consists in providing explicit examples of transcendental continued fractions. The main purpose of the present work is...

Source: http://arxiv.org/abs/math/0511682v1

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Jul 24, 2013
07/13

by
Yann Bugeaud; Stephen Harrap; Simon Kristensen; Sanju Velani

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Let A be an n by m matrix with real entries. Consider the set Bad_A of x \in [0,1)^n for which there exists a constant c(x)>0 such that for any q \in Z^m the distance between x and the point {Aq} is at least c(x) |q|^{-m/n}. It is shown that the intersection of Bad_A with any suitably regular fractal set is of maximal Hausdorff dimension. The linear form systems investigated in this paper are natural extensions of irrational rotations of the circle. Even in the latter one-dimensional...

Source: http://arxiv.org/abs/0807.3863v2